Introduction
A Geometric Progression (G.P.) is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed non-zero number called the common ratio (r). Understanding the nth term formula of a G.P. helps in finding any specific term in the sequence without listing all preceding ones.
Pattern: Geometric Progression (G.P.) – nth Term
Pattern
The nth term (Tₙ) of a geometric progression is given by:
Tₙ = a × rⁿ⁻¹
Here, a = first term, r = common ratio, and n = number of the term.
Step-by-Step Example
Question
Find the 7th term of the G.P. 2, 6, 18, 54, …
Solution
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Step 1: Identify a, r, and n
First term a = 2. Common ratio r = 6 ÷ 2 = 3. Required term n = 7.
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Step 2: Apply the nth term formula
Tₙ = a × rⁿ⁻¹ = 2 × 3⁶ = 2 × 729 = 1458.
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Final Answer:
The 7th term of the G.P. is 1458.
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Quick Check:
6th term = 486, multiplying by r = 3 → 486 × 3 = 1458 ✅
Quick Variations
1. Common ratio r can be a fraction (decreasing G.P.).
2. When terms alternate in sign, r is negative.
3. If any two terms are known, you can find r using r = (T₂ / T₁) or r = (Tₙ / Tₘ)^(1/(n-m)).
Trick to Always Use
- Step 1: Always find r first using any two consecutive terms.
- Step 2: Substitute in Tₙ = a × rⁿ⁻¹ carefully - exponent mistakes are common.
Summary
Summary
In a Geometric Progression (G.P.):
- The ratio between consecutive terms is constant.
- The nth term formula is Tₙ = a × rⁿ⁻¹.
- If r > 1 → sequence increases rapidly; if 0 < r < 1 → sequence decreases.
- For alternating sign series, r is negative.
Hint: Multiply the first term by rⁿ⁻¹ directly.
Common Mistakes: Using n instead of (n-1) in the exponent.